Paper and Mathematica expressions.
Zvi Bern, John Joseph Carrasco, Wei-Ming Chen, Alex Edison, Henrik Johansson, Julio Parra-Martinez, Radu Roiban, Mao Zeng
We use the recently developed generalized double-copy construction to obtain an improved representation of the five-loop four-point integrand of N=8 supergravity whose leading ultraviolet behavior we analyze using state of the art loop-integral expansion and reduction methods. We find that the five-loop critical dimension where ultraviolet divergences first occur is Dc=24/5, corresponding to a D8R4 counterterm. This ultraviolet behavior stands in contrast to the cases of four-dimensional N=4 supergravity at three loops and N=5 supergravity at four loops whose improved ultraviolet behavior demonstrates enhanced cancellations beyond implications from standard-symmetry considerations. We express this Dc=24/5 divergence in terms of two relatively simple positive-definite integrals reminiscent of vacuum integrals, excluding any additional ultraviolet cancellations at this loop-order. We note nontrivial relations between the integrals describing this leading ultraviolet behavior and integrals describing lower-loop behavior. This observation suggests not only a path towards greatly simplifying future calculations at higher loops, but may even allow us to directly investigate ultraviolet behavior in terms of simplified integrals, avoiding the construction of complete integrands.